The Logical Puzzle - 25 Jan 2006

FEBRUARY'S PUZZLE: PIECE OF CAKE

This month's tasty puzzle is from Michael Rys, Microsoft program manager in charge of SQL Server XML technologies, who faced the brainteaser in his job interview: You have a cake shaped as a rectangle. The cake features a rectangle-shaped hole in an arbitrary position, as Figure A shows. With a knife, you need to cut the cake so that you end up with two equally sized portions—that is, you have the same amount of cake in each portion. Note that the cake has a layer of frosting on top, and the two pieces must also have the same amount of frosting.

SOLUTION TO JANUARY'S PUZZLE

In last month's logical puzzle, you were a contestant in a game show, standing in front of three curtains—one of which hid a car. You didn't know which curtain hid the car, but the host of the show did. You first had to choose one curtain, which would remain closed while the host opened one of the other curtains, which didn't hide a car. You then received a final chance to either stay with your original curtain choice or select the other closed curtain. Which action would give you a higher probability of winning the car?

Most people think that each choice—sticking to your original choice and selecting the other curtain—has the same probability of winning a car: 50 percent. However, keep in mind that you made your first choice independent of any other action or knowledge and that, regardless of your original choice, the host will always open a curtain that doesn't hide a car. Sticking with your original choice remains independent of the event that follows, so you have a one-third probability of winning whether you select curtain 1 and stay with that choice, curtain 2 and stay with that choice, or curtain 3 and stay with that choice.

In contrast, the host's choice of which curtain to open is dependent on your original choice. And changing your original choice, in turn, is dependent on his choice. Thus, changing your original choice would be based on new knowledge. Given that only two options remain after the host opens one of the curtains, the probabilities of winning if you stick to your original choice or move must add up to 1 (100 percent). So, the probability of winning if you select the other curtain is 1 ? 1/3 = 2/3. Statistically, two out of three people who select the other curtain will win the car, while only one out of three who stick with their original choice will win.

If you're still not convinced, I suggest one of two things:

Write a program that simulates the game, then statistically check the results.

Share the puzzle with statisticians or mathematicians, and see what they have to say.